We study the Mumford–Tate conjecture for hyperkähler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O’Grady’s ten dimensional example, and all of their self-products. For an arbitrary hyperkähler variety whose second Betti number is not 3, we prove the Mumford–Tate conjecture in every codimension under the assumption that the Künneth components in even degree of its André motive are abelian. Our results extend a theorem of André.

中文翻译：

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关于hyperkähler变种的Mumford-Tate猜想

我们研究了hyperkähler变种的Mumford-Tate猜想。我们证明，对于所有变体，完全猜想都成立，这些变体等同于K3曲面上的希尔伯特点方案或O'Grady的十维实例及其所有自积。对于第二贝蒂数不为3的任意hyperkähler变体，我们假设在其偶数André动机中的Künneth分量是阿贝尔的前提下，证明了每个余维中的Mumford-Tate猜想。我们的结果扩展了安德烈定理。